ME 475/675 Introduction to
Combustion
Lecture 27
Variable area and pressure equations, Problem 6.11
Announcements
• Midterm 2, November 13, 2015 (two weeks from Friday)
• HW 8b, Due now, Example 6.2 Numerical solution
• HW 9 Due now, Problem X5
• HW 10 Due Monday Nov ??, 2015
• Friday, 10/30/2015, Holiday
• Opportunities at Oak Ridge National Labs (Tennesee)
• hands-on research in a-real world setting with award winning scientists
• Visit: http://www.orau.org/ornl[orau.org]
• Or contact: ORNL Education Programs at ornledu@orau.org, or Leslie Fox at (865)
576-3427
Broader Impact Assignment
• Two important ABET Student Learning Objectives:
• Students will demonstrate :
• A recognition of a need for, and an ability to engage in, life long learning (graduate school, continuing
education, short courses, technical training, self instruction by reading articles or textbook)
• A knowledge of contemporary issues
• Two choices, Both due November 6, 2015, two paragraph summaries
• Seminar: Used Nuclear Fuel: Storage, Transportation, and Disposal – Technical, Political and
Other Issues
• John Wagner, Director, Reactor & Nuclear Systems Division, Oak Ridge National Laboratory
• Noon, November 2, 2015, DMS 102
• Hosted by UNR American Nuclear Society’s Student Chapter
• President: Kodi Summers klsummers92@gmail.com
• Article: Dependence of Fire Time of Concern on Location of a One-assembly Transport
Packages
Plug-Flow Reactors
• Assumptions
• Quasi-one dimensional
𝑑
• Steady state,
=0
𝑑𝑡
• No-viscosity 𝜇 = 0
• Axial turbulent and molecular diffusion is small
compared to advection (high enough axial velocity)
• The quantities are ≠ 𝑓𝑛(𝑟, 𝜃)
• If velocity varies, then so does pressure
• Integrate to find 𝑇 𝑥 , 𝑌𝑖 𝑥 , 𝜌 𝑥
• At each location also need to calculate
𝑚
𝜌 𝑥 𝐴(𝑥)
𝜌 𝑥 𝑅𝑢 𝑇(𝑥)
𝑀𝑊𝑚𝑖𝑥
• 𝑣𝑥 𝑥 =
• 𝑃 𝑥 =
• Like the transient constant-pressure reactor, but varies
with location instead of time.
𝑌𝑖 𝑥
What do we expect?
•
General Plug-Flow Reactor
• What’s different from problem X3?
• Area A x ,
𝑑𝐴
𝑑𝑥
≠0
• Flow kinetic energy is not small compared to enthalpy
•
𝑣𝑥2
2
≪ℎ
• Pressure 𝑃 is not necessarily constant
• Species can have different, temperature-dependent properties
Conservation Laws
• Mass
• 𝑚 = 𝜌𝑣𝑥 𝐴
•
𝑑
𝑑𝑥
𝜌𝑣𝑥 𝐴 = 0
• Momentum
•
𝑑𝑃
𝑑𝑥
+
𝑑𝑣𝑥
𝜌𝑣𝑥
𝑑𝑥
=0
• Energy (including kinetic, here 𝑄 " is from flow)
•
𝑑
𝑣2
ℎ+ 𝑥
2
𝑑𝑥
+
𝑄" 𝒫
𝑚
=0
• Species
•
𝑑𝑌𝑖
𝑑𝑥
=
𝜔𝑖 𝑀𝑊𝑖 𝐴
,𝑖
𝜌𝑣𝑥
= 1,2, … , 𝑀
Manipulate, combine and solve ….
• Use
• 𝜔𝑖 = 𝑓𝑛 𝑌𝑖 , 𝑇, 𝑃 ; 𝑃 = 𝜌𝑅𝑇; 𝑅 =
𝑅𝑢
1
;
𝑀𝑊𝑚𝑖𝑥 𝑀𝑊𝑚𝑖𝑥
=
𝑌𝑖
; 𝑣𝑥
𝑀𝑊𝑖
=
𝑚
; 𝑐𝑃
𝜌𝐴
=
𝑌𝑖 𝑐𝑃𝑖
• Need 𝑌𝑖 (𝑥), 𝜌 𝑥 and 𝑇 𝑥
• Assume 𝑄 " 𝑥 , 𝐴 𝑥 and 𝑚 are given
• Find … (page 209)
•
•
•
𝑑𝑌𝑖
𝑑𝑥
𝑑𝜌
𝑑𝑥
𝑑𝑇
𝑑𝑥
=
=
𝜔𝑖 𝑀𝑊𝑖 𝐴
,𝑖
𝜌𝑣𝑥
= 1,2, … , 𝑀 (eqn. 6.53)
𝑅𝑢
1−
𝑐𝑃 𝑀𝑊𝑚𝑖𝑥
1 𝑑𝐴
𝜌2 𝑣𝑥2
𝐴 𝑑𝑥
𝜌𝑅𝑢
+
𝑣𝑥 𝑐𝑃 𝑀𝑊𝑚𝑖𝑥
𝑃
=
𝑣𝑥2 𝑑𝜌
𝜌𝑐𝑃 𝑑𝑥
𝑣𝑥2 𝑑𝐴
+
𝑐𝑃 𝐴 𝑑𝑥
−
𝜔𝑖 𝑀𝑊𝑖 ℎ𝑖
𝜌𝑣𝑥 𝑐𝑃
𝑣2
1+ 𝑥
𝑐𝑃 𝑇
−
𝑄" 𝒫
𝑚𝑐𝑃
𝑀𝑊𝑚𝑖𝑥
𝑀𝑊𝑖 𝜔𝑖 ℎ𝑖 −
𝑐 𝑇
𝑀𝑊𝑖 𝑃
𝜌𝑅𝑢 𝑄" 𝒫
+
𝑣𝑥 𝐴𝑐𝑃 𝑀𝑊𝑚𝑖𝑥
−𝜌𝑣𝑥2
(Eqn. 6.51 is for 𝑄" = 0)
"
(Eqn. 6.52 is for 𝑄 " = 𝑄𝑜𝑢𝑡
= 0)
Problem 6.11
Extra credit-turn in first lines of Excel next meeting
Homework-turn in numerical solution
• Develop a plug-flow-reactor model using the same chemistry and
thermodynamics as in Example 6.1. Assume the reactor is adiabatic. Use the
model to:
A. Determine the mass flow rate such that the reaction is 99 percent complete in
a flow length of 10 cm for 𝑇𝑖𝑛 = 1000𝐾, 𝑃𝑖𝑛 = 0.2 𝑎𝑡𝑚, and Φ𝑖𝑛 = 0.2. The
circular duct has a diameter of 3 cm.
B. Explore the effects of 𝑇𝑖𝑛 , 𝑃𝑖𝑛 , and Φ𝑖𝑛 on the flow length required for 99
percent complete combustion using the flow rate determined in Part A.
• Does this reactor operate at constant volume, at constant pressure, as a wellstirred, or as a plug-flow?
Example 6.1 and 6.2 Chemistry
Example 6.2
• Species Production
• 𝜔𝐹𝑢𝑒𝑙 = −6.186 ∗ 109 𝑒𝑥𝑝
• 𝜔𝐹𝑢𝑒𝑙 = −6.186 ∗
• 𝜔𝑂𝑥 =
𝐴
𝐹 𝑆𝑇
−15098𝐾
109 𝑒𝑥𝑝
𝑇
𝐹𝑢
𝑌𝐹𝑢
0.1
0.1
𝑂2
1.65
0.233𝑌𝑂𝑥
1.65
𝑃 1.75
𝑅𝑢 𝑇
𝜔𝐹𝑢𝑒𝑙
𝐴
𝐹 𝑆𝑇
• 𝜔𝑃𝑟𝑜𝑑 = −
−15098𝐾
𝑇
+ 1 𝜔𝐹𝑢𝑒𝑙
• Inlet Conditions for Problem 6.11
• 𝑌𝐹,𝑖𝑛 =
1
1+
𝐴
/Φ
𝐹 𝑆𝑇
• 𝑇𝑖𝑛 = 1000𝐾
• Specify:
•𝑚
• 𝑄" = 0
; 𝑌𝑂𝑥,𝑖𝑛 =
1
1+Φ/
𝐴
𝐹 𝑆𝑇
; 𝑌𝑃𝑟,𝑖𝑛 = 0
Problem 6.11
Excel Solution Method
• Starting Point
• http://wolfweb.unr.edu/homepage/greiner/teaching/MECH.475.675.Combustion/Prob.6.11.start.xlsx
• Pay attention to
• Integration step size
• Avoiding raising negative numbers to a non-integer power
mdot
phi
dx
x
Yfuel
kg/s
m
m
kg/kg
0.00125
1
0.0001
0 0.058824
Vx
Wf
d[Yfuel]/dx d[Yox]/dx
m/s
kmole/m3s
kg/kgm
kg/kgm
25.02663
-0.0028245 -0.0463193 -0.74110867
Yox
Ypr
rho
T
P
kg/kg
kg/kg
kg/m3
K
kPa
0.941176
0 0.07066
1000
20.26
d[Ypr]/dx
d[rho]/dx
dT/dx
%FuelRemaining
kg/kgm
kg/m4
K/m
0.78742796 -0.109279439 1543.169182
1
Turn this in for Extra Credit:
mdot
kg/s
0.00125
phi
1
dx
m
0.0001
x
m
Yfuel
Yox
kg/kg
kg/kg
0 0.058824 0.941176
Ypr
kg/kg
0
rho
kg/m3
0.07066
T
K
1000
P
Vx
Wf
d[Yfuel]/dx d[Yox]/dx
kPa
m/s
kmole/m3s
kg/kgm
kg/kgm
20.26 25.02663
-0.0028245 -0.0463193 -0.74110867
d[Ypr]/dx
d[rho]/dx
dT/dx
%FuelRemaining
kg/kgm
kg/m4
K/m
0.78742796 -0.109279439 1543.169182
1